Dr. Esam hussec:oin ..:.....4 ..:.....Mo=n=te::.......=Ca=I=·l0:c...-::...Pa==-rticle Transport with MCNP

CHAPTER 2: THE MONTE CARLO METHOD

The Monte Carlo method is a method of approximately solving mathematical andphysical problems by the simulation of random quantities. The terminology"Monte Carlo" comes from the city of Monte Carlo in the principality of Monaco,famous for its gambling houses.

The computational algorithm is relatively simple in Monte Carlo calculations. Thealgorithm consists, in general, of a process for producing a random event. Theprocess is repeated N times, each trial being independent of the others, and theresults of all trials are averaged together to provide an estimate of the quantity ofinterest.The process is similar to performing a scientific experiment and is sometimescalled the method of stochastic. or statistical experiments or trials.

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The error associated with the estimated quantity is, as a rule, inverselyproportional to the square root of the number of trials. that is

1/2error ex (1/N)

It is clear that, to decrease the error by a factor of 10 (in order to obtain anothersignificant digit in the result), it is necessary to increase N (and thecomputational effort) by a factor of 100. To attain high precision in Monte Carlocalculations is clearly impossible.

The Monte Carlo method is most effective in soiving problems in which theresults need to be accurate to less than a few percent. It is important to point outhere however that, unlike other deterministic methods, the Monte Cario methodprovides an answer with an error associated with it, so that a confidence level inthe result can be established.

The main advantage of the Monte Carlo method is its ability to handle complexgeometry. Its main limitation is that it only provides solutions at specifiedlocations, unlike deterministic methods which provide solutions at all points inthe space considered.

Oeoarlment of Mechanical Enaineerina Universitv of New Brunswick Fredericton. N. B. Canada

Dr. Esam dusseill 6 Monte Carlo Particle Transport with MCNP

Since the Monte Carlo method is a computat:onal process in which randomvariables are used, we begin by explaining what it is meant by a random variableand reviewing some important statistical concepts.

2.1 Random Variables

In ordinary English usage, a random variable is the outcome of any process thatproceeds without any discernible aim or direction.

Mathematically the word random variable means that we do not know the valueof a particular quantity in any given case, but we know what values it canassume and we know the probabilities with which it assumes these values. Thena random variable, X, is defined discretely by the table

where the xIs are possible values of X and the PIs are the correspondingprobabilities. Then one writes

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For continuous random variables, a function p(x) in some interval (a,b) isassigned and called the probability density function (pdf), or the densitydistribution, such that

b

Pea < X < b)= f p(x')dx' such that; p(x) > 0a

The zero-th moment of this function is normalized such that

00

f p(x)dx =1-00

The first moment of the distribution provides the so-called expected value ormathematical expectation

b

E(X)= f xp(x)dxa

The second central moment defines the variance of the distribution

b

cl(X)= f [x-E(X)!p(x)dxa

Oeoanment of Mechanical Enoineerino Universitv of New BrunswicK Fredericton. N. B. Canada

=Dr~.~E=sa=m~H=US=S=e"=D =8 ~M=o=nt=e~C=a=r=10 Particle Transport with MCNP

The cumulative density function, cdf is defined as

Xo

F(xo)=P(X<xo)= f p(x)dxa

Then,

b'pea' < X < b') = f p(x)dx =F(b')-F(a')

a'

The cumulative density function represents an area under the pdf extending froma to xo'

The cdf is particularly useful in Monte Carlo calcuiations as shown later.

Dr. Esam Hussein

2.2 Abstract Analysis

9 MOrita Carlo Particle Transport with MCNP

An event means the occurrence of a specified outcome of an experiment. LetQ be the event that includes all possible events. Then any event A, Q, that is Ais contained in Q.

The probability is a real-valued function of the events of an experiment satisfying

P(0)=O, P(Q)=1; 0 <P(A)<1; for all A c n

00 00

P(U Ai)= L P(Ai); if Ai n Aj = 0, i:;rj1=1 1=1

where u reads union and signifies the fact that the event exists simultaneouslyin the spaces considered, while n reads intersection and defines eventscommon to the concerned spaces.

The above abstract notion of probability is more general than the frequencynotion usually used in statistical analysis. In the frequency probability analysis, inan experiment repeated n times, with an event A occurring nA times, oneexpects (nA)/n to cluster about a unique number P(A).

Deoar1ment of Mechanical Enoineerino Universitv of New Brunswick Fredericton. N.B. Canada

Dr. Esam Hussein 10 Monte Carlo Particle Transporc with MCNP

The abstract notion of probability requires however only that the function Passigns to every event A a number with the above probability.

The probability is a real-valued function in certain subsets of G, which we callthe event of Q. Certain real-valued functions of the points of Q are calledrandom variables.

Let a paint of Q denoted by co and let ~ be a real-valued fUllction on the points ofQ. Let

A(t)={ (j)1~«(j)) < t}

This defines the set of all points (j) such that ~«(j)) < t. Then A (t) is a subset of Qwhich depends on the real number t.

Dr. Esam Hussein 11 Monte Carlo Particle Transport with MCNP

If for every t, the set A (t) is an event, then ~ is called a random variable. Then

P{A(t)}= P{(j)I~((j)) < t}=pfEo < t}; defined for every t

The above is the formal definition of random variables.

The real-valued function of a real variable defined by

F(t)= pfEo < t}

is called the distribution function or the cumulative density function.It has the following characteristics

1. F is continuous on the right at every event t

2. F is a monotone non-decreasing function

3. F (-00)=0 and F (00)=1

4. F (a)-F(b)=P(a < x < b), for a<b

5. If to is a point of discontinuity of F with a jump of height p, then p{~ =to} Pand there is a non-zero probability that the random variable takes on the valueto·

Deaartment Of Mechanical Enoineerino Universitv of New Brunsw/C!; Fredericton. N. B. r;anada

Dr. Esam Hussein 12 Monte Carlo ?article Transport with ~1CNP

6. If the derivative of F with respect to t exists at point t, then

Iimt.--+O P{t-N2 < x <t + #2) = (dF/dt)L1 = f(t)

If the derivative, f(t) exists, it is called the probability density function, or simplythe density function.

If N independent trials of an experiment are performed, the probabilityspace 0 consisting of all N-tuples (ro 1 '00' Q)N) of points of 0 is

Define N random variables ~i on ON by

Dr. Esam Hussein 13 Monte Carlo Pafticle Transport with MCNP

If ~ is a discrete random function of N variables, such that

~(N) = ~ ~i1=1

is also a random variable on QN and represents the total number of occurrencesof the event co in N repetitions of the experiment.

2.2.1 Tchebycheff Theorem

This theorem states mathematically that for any random variable, ~,

of any distribution function with a mean, or expected value m, and a standarddeviation ()

P~~-ml > k a}< 1/k2

, k> 0If we use k to define an "error" E=k cr, then for the random variable ~(N)

p~ ~(N) -pI> E} < 1/(4 Nl)

This theorem reiterates the fact represented by equation (2.1) {error u::. (1/N//2}

that to reduce the error by a factor of two, the number of trials of the experimentmust be quadrupled.

Deoartmer.t of Mechanical Enaineerina Universitv cf New Brunswicx Fredericton. N.B. Canada

-cDr~.-=E~sa__m-cHU~SS~e__in 14 M-c0__nt~e Carlo Particle Transport wlth MCNP

2.2.2 Central Limit Theorem

This is also called the law of large numbers and states essentially that ~(N) (willbe approximately normally distributed even if ~ is not.

The theorem states formally that if ~1' .. ~N is a sequence of independent andidentically distributed random variables with a common mean m and variance (l,then

_ N

~ =(1/N) L ~ I1=1

is asymptotically normal (m, crIN1/2

), that is

- 1/2 1r x 2limN-faJP{(~ -.71)/( crlN )}<x} =(1/(2 1[ L)} J exp(-t /2) dt

-00

The theorem assumes that both m and cr exist, that is they are given byabsolutely convergent integrals. Applying the Tchebycheffs theorem, then

P{I{(~ -m)/( crIN1/2)}I< E '} leads to (1/(2 1[1/2)} {1' exp(-l/2) dt- 7 exp(-l/2)dt}

-00 g'

Dr. Esam HLI~sein 15 Monte Cario Particle Transport with MCNP

In the above equation leads to implies that they are asymptotically equal.

d(cr/.,fNl 2P{I ~-ml< £ }={2/ ;r1/2} J exp(-r /2)dt

o

The central-limit theorem is the backbone of the Monte Carlo method. Theaverage value ~ is used as an estimate of the random variable ~. This valueapproaches the true expected value, m, as the number of trials approach infinity.

The variability estimated by

2 N ') N 7-s =(1/N) L ~~I - {(1/N) L ~i}

1=1 1=1

is called the sample variance. It is not directly an estimate of the distributionvariance. It can be stated however that

The confidence interval in the estimated value of ~ can be defined by- - 1~

[~+cre , ~-cre]' where (Je=cr/N

Deoartment of Mechanical Enaineerina Universitv of New Brunswick Fredericton. N.B. Canada

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Since (J is not known, the following estimate is used

(Je2=(J2/N={1/(N-1)}[(1iN) £ ~2 -{(1iN) £ ~i}2]

1=1 1=1

A useful quantity used in Monte Carlo computations is called the fractionstandard deviation, fsd, defined as

fsd= (J/~

An fsd of less than 0.05, or 5%, is usually required in Monte Carlo calculations.

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2.3 Construction of Samples

The realization of a random variable, ~i from a pdf f(x) is obtained byconstructing a sequence of numbers t1, ... ,tn' such that

b

P{a < t <b} =f f(x) dxa

and

b

P{a < (1' ... , (n < b) =P{a < (1< b) ... P{a < t;n< b}=[J f(x)dxf, , ,a

The above equation implies that the random variables ~i1' .. , ~in' are mutuallyindependent, if t

il' 0' ti n' are all different. ", ,

The sequence of random numbers, P1' ... : Pn, such that a< Pi 51, representssamples drawn independently from a uniform pdf in the interval [0,1]. that is

b

P{a < Pi <b) = J 1 dx = b-aa

andn

P{a < Pi1' ... , Pin<b} = (b-a) , i1, ..., in' are all differentDeoartment of Mechanical Enoinf'erino Universitv of New Brunswick Fredericton. N.B. Canada

Dr. Esam Hussein 18 Monte Carlo Particle Transport with MCNP

Now by setting a sampled random number p equal to the cdf r(x), that is

x

F(x)=P{~< x}= f f(t)dt = pa

one can solve for x, and consequently obtain a value that is sampled from thedistribution f(x).

For a discrete distribution, one constructs the edf

F(x)= I: Pi for X<Xi

where Pi is the probability of occurrence of XI. The sampling of an xj is thenachieved as follows

j-1 j

I: Pi < p< I: Pi1=1 1=1

where p is a random number uniformly distributed in the interval (0,1).

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2.4 Random Number Generation

Digital random number generators are nowadays a standard feature in almostall computer systems. The generated numbers are called pseudo randomnumbers as they are not purely random. They must satisfy however twoimportant criteria:

Equi··distribution: each number has the same probability of occurrenceas any other number in the set.

Independence: the occurrence of any given number should not depend on theprevious occurrence or any subsequent occurrence of any other number.

The modulus method is perhaps the most widely used method. Given anyconstant a the random numbers are generated as follows

Pj=a Pi-1 (mod M)

where M=2k, and k is the number of bits per word in the computerbeing used. Modulus is a number or quantity that produces the same remainderwhen divided into each of two quantities. A = B (mod M) reads A iscongruent to B module M and means A is the remainder of B/M1

.

I For example: 15 (mod 13)=2Oeoarfment of Mechanical Enaineerina Universitv of New Brunswick Fredericton. N.B. Canada

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2.5 Monte Carlo Simulation of Particle Transport

2.5.1 Essential Requirements

SourceThe position, geometry, directional distribution and energy distribution of thesource must be specified. In transient analysis, the change of the source withtime must be also known. Fission sources and collision sources are determinedby the cross section of the material and need not be specified as inputparameters. The fission d!stribution with energy, X(E), must however bespecified, in order to determine the energy of the emerging neutron. A sourceparticle is usually assigned a statistical weight, W, the significance of which isexamined later.

GeometryThe Monte Carlo method can handle complex geometries. The geometry musthowever be specified in such a way that enables tracking of the particlethroughout the system and relating the position of the particle within the systemto the material, or more specifically material cross section.The geometry can be specified via analytical geometry procedures, which definethe surfaces of different geometrical objects. Alternatively, the geometry may bespecified via a set of elementary bodies, combined together using logicaloperators to form a zone of a particUlar material. This is called the combinatorialgeometry method and is utilized in the MORSE code.

_Or_,E_s_am_l-I_u_SSe_in ----.:2::..:..1 Monte Carlo Particle Transport with MCNP

Material Cross SectionsThe cross sections for the different materials encountered must be supplied as afunction of energy. The Legendre expansion coefficients for each material arealso needed, if an anisotropic scattering is considered.

The cross sections are processed prior to the simulation to prov!de theprobability table, which determines the distance to be travelled by the particleuntil the next collision, the outcome of the collision, and the outgoing energy andangle of a scattering event; in addition to the number of neutrons per fission forfissile materials.

ScoringThe scoring process is determined by a variety of estimators which evaluate thefluence, or fluence-like quantities, at a point or a region. Statistical estimates,including the average and the variance of the average are estimated at the endof the random walk process.

The surface crossing estimator evaluates the flux crossing a surface, byaccumulating the weight of particles crossing the surfaces divided by theabsolute value of the cosine of the angle between the normal to the surface anddirection of the incident particle. Provisions are made to avoid small anglescosines.

Deoartment of Mechanical Enoineerino Universitv of New Brunswick Fredericton, N,B. Canada

_Dr__.__E__sa__m~H~us~sec-in 2__2 Monte Carlo Particle Transport with MCNP

The track length estimator evaluates the fluence by summing the track length ofparticles crossing a given zone, divided by the volume of the zone.This is usually suitable for evaluating the fluence in void or air regions, andregions containing a low density material.

The collision density estimator adds up the weight of particles colliding within azone, divided by the total cross section of the material and the volume of thezone. The estimator provides adequate estimates for the fluence in regions ofhigh density materials, where a large number of collisions are anticipated.

In all the above estimators, the particle must visit the region, or surface ofinterest. In situations where ttle probability of the particles reaching the region ofinterest is low, indirect estimates, called statistical estimation are used. Theseestimators evaluate the probability of the next collision being at the detector site.This called the next event estimator and is particularly useful for point detectors,where there is only one possible position for the "next collision". Note that theparticle being tracked does not alter its original position, only the probability ofthe next collision being at the detector site is evaluated and stored.

2.5.2 Example

In order to illustrate the above points, let us consider the relatively simpleproblem of evaluating the fluence through a shielding slab, wittl a neutron sourceon one side and a detector on the other side.

_Or_,E_s_am_H_u,_'se_in ----'2:::.3 M:.::.o::..:-n=te Car 10 Particl8 Transport with MCNP

Source ParametersIf we assume a point, mono-energetic and isotropic source, then only thedirection of the incident particle need to be sampled, ill a steady state problem.Since we are not interested in neutrons directed away from the target, theangular probability becomes

p(Q)dU= d 0./2 rc =(sin8d8 dcp)/2rc=(dcos8dcp)(2n:)

where 6 is the polar angle and $ is the azimuthal angle.

Equating the cumulative probability for cos 8 to a random number P1 sampledfrom a uniform distribution in the interval (0,1), then

cos El

P1== J (1/2) dcos 8'=(cosO +1)/2-1

The inversion of the above leads to an equation for selecting 8

8 =cos -1 (2 P1 -1)

Similarly, the angle <p is sampled from the relationship <P=rcP2 where P2 is anotherrandom number.

Oeoartment of Mochanical Enaineerina Universitv of New Brunswick Fredericton. N.B. Canada

Dr. Esam Hussein 24 Monte Carlo Particle Transport with MCNP

Distance of TravelNext, one needs to determine the distance the neutron will travel until it collides.The probability of a neutron experiencing its first interaction between thedistances x and x+dx is equal to Lt exp(-2: x), where Lt is the total cross sectionof the material encountered.

As shown earlier, equation (2.44) {X=-(1/L) In p} provides the method forsampling the distance, x, the neutron will travel until it collides.

3.2.3 Type of InteractionThe type of interaction which takes place at the position defined by the distancex is determined by the so-called activation cross sections, which are usuallyLscatter, v Lfission, and 2:capture. In certain circumstances, the cross section ofsome particular reactions, such as the (n,p) reaction, may be specified.The activation table of probabilities is converted into a cumulative probabilitytable, enabling the selection of the proper interaction.

If the interaction is determined to be an absorption process, the random walk ofthe particle may be terminated. This is called analog Monte Carlo and is notoften used as it may result in early termination of the random walk. Alternatively,a non-analog process is used in which the partide weight is reduced by the non­absorption probability, (Ltotal-Lcapture)/ Ltotal. and a particle scattering orfission is sampled.

Dr. Esam Hussein 25 Monte Carlo Particle Transport with MCNP

This process allows the particle to fully complete its path within the system, untilit escapes the system or is terminated by a weight cut-off, or an energy cut-off,or some other pre-specified process.

3.2.4 Energy of Outgoing ParticleIn a non-fissile material, the only interaction possible in a non-analog MonteCarlo is particle scattering. One needs then to determine the energy and angleof the particle emerging from the collision.

Let us assume an elastic isotropic neutron scattering process.Then, the energy of the outgoing particle can lie any where from the energyof the incident particle Ei to the minimum possible energy a E, wherea=[(A-'1 )/(A+1 )]2, with A being the mass number of the element considered. Theprobability of the particle reaching an energy E is given by

p(E)dE=dE/[Ej(1-a)]

Equating the cumulative probability to some random number P3' oneobtains

E=P3 El1-a) + a E,

Deoattmem of Mechanical Enaineerina Universitv of New Brunswick Fredericton. N.B. Canada

Dr. Esam Hussein 26 Monte Carlo Particle Transport with MCNP

Problem. Devise a method for determining the outgoing energy for Isotropicscattering in a chemical compound such as water.

The outgoing energy is sampled from the above equation. Since isotropicscattering is assumed, the outgoing direction can be sampled using a proceduresimilar to that used for the source, except that the whole 4 nof the azimuthalangle must be considered.

Once the direction and energy of the scattered particle are determined, thedistance of flight until the next collision is evaluated, and so on. Note, however,in this one-dimensional problem it is sufficient to determine the x position of thecollision site, as x=xi+dcos 8 sin cp, where Xi is the initial position of the particleand d is the distance the particle travels between collisions.

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ScoringA simple scoring process is to employ the boundary crossing estimator at theboundary far away from the source. For a deep penetration problem, e.g. thickshield, the probability of the particle crossing the shielding slab is very low.Some so-called biasing or importance sampling techniques can be employed.These techniques involve splitting which increases the number of particlestravelling towards the location of interest (forward in the problem considered),and Russian roulette which kills most of the particles travelling in the "wrong"direction. The exponential transformation technique may also be employed. Inthis technique, the total cross section is artificially decreased, to enable theparticle path length between collisions to stretch, and consequently be able tocross the slab. In all these biasing techniques, the palticle weight is adjustedsuch that the resulting estimates are unbiased.

2.6 Work Problems1. Prove that the cr2(X)=E[{ X-E(X)}2]2. Show that the following procedure represents sampling from the distributionL exp(-L x), where 22 is a constant

X=-(1/L) In p

Oeoartment of Mechanical Enoineerino Umversitv of New Brunswick Fredericton. N.B. Canada